Finite Group Databasebeta

← D5×C4⋊#226 #14512805

Subgroups — D5×C4⋊#226 #14512805

Order 1920.

550 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 71 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D5×#226C212yesmaximal
2C4×D5×SL(2,𝔽3)C211yesmaximal, characteristic
3C4⋊(C5⋊#226)C211yesmaximal, characteristic, complemented
4C5×C4⋊#226C211yesmaximal, characteristic, complemented
5#11751597C212yesmaximal, complemented
6D5×#462—31nomaximal, Hall
7C2×D5×SL(2,𝔽3)C2211yescharacteristic
8C2×C5⋊#226C2212yes—
9C2×C5×#226C2212yes—
10D5×#226—24no—
11C4×C5×SL(2,𝔽3)C2211yescharacteristic
12D5×C4⋊Dic3—41nomaximal
13Dic5×SL(2,𝔽3)C2211yescharacteristic
14C2×C4⋊#226—51nomaximal, Hall
15C2×D5×Q16—32no—
16D5×C4⋊Q8—31no—
17C4×Q8×D5D311yescharacteristic
18C5×#462—31no—
19#65471—32no—
20#65379—31no—
21#65320—31no—
22#65138—31no—
23D5×C4⋊C8—31no—
24D5×#103—32no—
25#65082—32no—
26C2×C3×C4×D5—41no—
27C2×C5×SL(2,𝔽3)C2311yescharacteristic
28C2×D5×Dic3—42no—
29D5×SL(2,𝔽3)D412yes—
30C5⋊#226—22no—
31C5×#226—22no—
32C4⋊Dic15—41no—
33C5×C4⋊Dic3—41no—
34#6668—42no—
35C22×#226—52no—
36C2×C4×SL(2,𝔽3)—51no—
37C4⋊#226—52no—
38C4⋊#226—51no—
39C4⋊#226D511yescharacteristic, direct product factor
40C2×Q8×D5—32no—
41C2×Q8×D5C2×D311yescharacteristic
42C2×C5×Q16—32no—
43C5×C4⋊Q8—31no—
44C4×C5×Q8C2×D311yescharacteristic
45Q8×Dic5C2×D311yescharacteristic
46#4090—32no—
47C2×#689—32no—
48C2×#689—32no—
49D5×Q16—64no—
50D5×Q16—64no—
51C2×Dic20—32no—
52C2×C8×D5—32no—
53D5×C4⋊C4—31no—
54D5×C4⋊C4—61no—
55D5×C4⋊C4—61no—
56#4034—61no—
57#4034—61no—
58#4033—31no—
59C42×D5—31no—
60#4015—31no—
61C4×Dic10—31no—
62C5×C4⋊C8—31no—
63C5×#103—32no—
64#3967—32no—
65#3948—32no—
66#3946—32no—
67#3942—32no—
68C4⋊#168—31no—
69C2×#462—151noSylow
70C22×C3×D5—41no—
71C2×C3×C4×C5—41no—
72C2×Dic15—42no—
73C2×C5×Dic3—42no—
74C2×C3×Dic5—41no—
75C3×C4×D5—44no—
76C5×SL(2,𝔽3)C2×D411yescharacteristic
77D5×Dic3—84no—
78C22×SL(2,𝔽3)—51no—
79C2×#226—52no—
80C2×#226—104no—
81C2×#226C2×D512yes—
82C2×C4⋊Dic3—201no—
83C4×SL(2,𝔽3)—51no—
84C4×SL(2,𝔽3)C2×D511yescharacteristic
85C2×C5×Q8—32no—
86C2×C5×Q8C22×D311yescharacteristic
87Q8×D5—32no—
88Q8×D5—64no—
89Q8×D5—64no—
90Q8×D5D1212yes—
91C2×C4×D5—31no—
92C2×C4×D5—31no—
93C2×C4×D5—62no—
94C2×C4×D5S411yescharacteristic
95C2×Dic10—31no—
96C2×Dic10—32no—
97C2×Dic10—62no—
98C5×Q16—62no—
99C5×Q16—62no—
100C2×C5×C8—32no—
101C5×C4⋊C4—31no—
102C5×C4⋊C4—61no—
103C5×C4⋊C4—61no—
104C42×C5—31no—
105#689—64no—
106#689—62no—
107#689—62no—
108C4⋊Dic5—31no—
109C4⋊Dic5—61no—
110C4⋊Dic5—61no—
111#683—61no—
112#683—62no—
113#683—62no—
114C4×Dic5—31no—
115C4×Dic5—31no—
116C4×Dic5—31no—
117C2×#168—32no—
118Dic20—64no—
119C8×D5—64no—
120C22×Q16—152no—
121C2×C4⋊Q8—151no—
122C2×C4×Q8—51no—
123#462—31no—
124#462—152no—
125#462—151no—
126#462—152no—
127#462—151no—
128#462—151no—
129C2×C4⋊C8—151no—
130C2×#103—152no—
131C22×C3×C5—41no—
132C2×C3×D5—42no—
133C2×C3×D5—42no—
134C2×C3×D5—42no—
135C3×C4×C5—42no—
136Dic15—82no—
137C3×Dic5—42no—
138C5×Dic3—82no—
139C22×C3×C4—201no—
140C22×Dic3—202no—
141C2×SL(2,𝔽3)—52no—
142C2×SL(2,𝔽3)C22×D511yescharacteristic
143#226—22no—
144#226—102no—
145C4⋊Dic3—41no—
146C4⋊Dic3—202no—
147C4⋊Dic3—201no—
148C22×D5C2×S411yescharacteristic
149C5×Q8—31no—
150C5×Q8—62no—
151C5×Q8—62no—
152C5×Q8C2×D1211yescharacteristic
153C2×C4×C5—31no—
154C2×C4×C5—31no—
155C2×C4×C5—62no—
156C2×C4×C5C2×S411yescharacteristic
157C2×Dic5—31no—
158C2×Dic5—31no—
159C2×Dic5—62no—
160C2×Dic5C2×S411yescharacteristic
161C4×D5—32no—
162C4×D5—32no—
163C4×D5—62no—
164C4×D5—124no—
165C4×D5#22614yescomplemented
166Dic10—32no—
167Dic10—61no—
168Dic10—62no—
169Dic10—62no—
170Dic10—124no—
171C5×C8—62no—
172#168—62no—
173C22×Q8—51no—
174C22×Q8—152no—
175C2×Q16—32no—
176C2×Q16—152no—
177C2×Q16—152no—
178C2×Q16—152no—
179C2×Q16—304no—
180C2×Q16—304no—
181C22×C8—152no—
182C4⋊Q8—31no—
183C4⋊Q8—151no—
184C4⋊Q8—152no—
185C4⋊Q8—301no—
186C4⋊Q8—301no—
187C4×Q8—51no—
188C4×Q8—151no—
189C4×Q8—151no—
190C4×Q8D3×D511yescharacteristic
191C2×C4⋊C4—151no—
192C2×C4⋊C4—301no—
193C2×C4⋊C4—301no—
194C2×C42—151no—
195C4⋊C8—31no—
196C4⋊C8—151no—
197C4⋊C8—152no—
198#103—32no—
199#103—152no—
200#103—152no—
201#103—152no—
202C2×C3×C5—41no—
203C2×C3×C5—41no—
204C2×C3×C5—41no—
205C3×D5—44no—
206C23×C3—201no—
207C2×C3×C4—41no—
208C2×C3×C4—204no—
209C2×C3×C4—201no—
210C2×Dic3—42no—
211C2×Dic3—202no—
212C2×Dic3—404no—
213SL(2,𝔽3)D4×D511yescharacteristic
214C22×C5C22×S411yescharacteristic
215C2×D5C4⋊S412yes—
216C2×D5C2×#22612yes—
217C2×D5#98412yes—
218C4×C5—31no—
219C4×C5—31no—
220C4×C5—61no—
221C4×C5—122no—
222C4×C5C2×#22612yescomplemented
223Dic5—32no—
224Dic5—61no—
225Dic5—122no—
226Dic5C2×#22612yescomplemented
227C2×Q8—32no—
228C2×Q8—52no—
229C2×Q8—151no—
230C2×Q8—152no—
231C2×Q8—152no—
232C2×Q8—302no—
233C2×Q8—304no—
234C2×Q8—304no—
235C2×Q8C2×D3×D511yescharacteristic
236C22×C4—51no—
237C22×C4—151no—
238C22×C4—151no—
239C22×C4—302no—
240Q16—62no—
241Q16—62no—
242Q16—304no—
243Q16—302no—
244Q16—304no—
245Q16—302no—
246C2×C8—32no—
247C2×C8—152no—
248C2×C8—304no—
249C4⋊C4—31no—
250C4⋊C4—61no—
251C4⋊C4—61no—
252C4⋊C4—151no—
253C4⋊C4—301no—
254C4⋊C4—301no—
255C4⋊C4—302no—
256C4⋊C4—302no—
257C4⋊C4—301no—
258C42—31no—
259C42—151no—
260C42—151no—
261C42—151no—
262C3×C5—41noHall
263C22×C3—41no—
264C22×C3—202no—
265C22×C3—202no—
266C22×C3—202no—
267C3×C4—42no—
268C3×C4—202no—
269Dic3—82no—
270Dic3—402no—
271C2×C5C2×C4⋊S411yescharacteristic
272C2×C5C22×#22611yescharacteristic
273C2×C5C2×#98411yescharacteristic
274D5C4⋊#22614yesdirect product factor
275C23—51no—
276Q8—31no—
277Q8—62no—
278Q8—62no—
279Q8—152no—
280Q8—301no—
281Q8—302no—
282Q8—302no—
283Q8—604no—
284Q8D5×D1211yescharacteristic
285C2×C4—31no—
286C2×C4—31no—
287C2×C4—51no—
288C2×C4—54no—
289C2×C4—62no—
290C2×C4—152no—
291C2×C4—152no—
292C2×C4—151no—
293C2×C4—151no—
294C2×C4—302no—
295C2×C4—302no—
296C2×C4—604no—
297C2×C4D5×S411yescharacteristic
298C8—62no—
299C8—302no—
300C2×C3—41no—
301C2×C3—41no—
302C2×C3—41no—
303C2×C3—204no—
304C5C2×C4⋊#22611yescharacteristic, complemented, Sylow
305C22—52no—
306C22—52no—
307C22—52no—
308C22C2×D5×S411yescharacteristic
309C4—31no—
310C4—31no—
311C4—52no—
312C4—61no—
313C4—122no—
314C4—152no—
315C4—301no—
316C4—602no—
317C4D5×#22612yescomplemented
318C3—41noSylow
319C2—54no—
320C2D5×C4⋊S411yescharacteristic
321C2C2×D5×#22611yescharacteristic
322C2D5×#98411yescharacteristic
subgroupG/H
center, FrattiniC22C2×D5×S4
commutator, cosocleC2×C5×SL(2,𝔽3)C23
FittingC4×C5×Q8C2×D3
socleC22×C5C22×S4
prime radicalC2×D5×SL(2,𝔽3)C22
hypercenterC2×C4D5×S4
hypocenterC5×SL(2,𝔽3)C2×D4
LCSG ▷ C2×C5×SL(2,𝔽3) ▷ C5×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C4⋊(C5⋊#226)C2semidirect
4. C5×C4⋊#226C2semidirect
5. #11751597C2semidirect
39. C4⋊#226D5direct (N × H)
165. C4×D5#226semidirect
222. C4×C5C2×#226semidirect
226. Dic5C2×#226semidirect
274. D5C4⋊#226direct (N × H)
304. C5C2×C4⋊#226semidirect
317. C4D5×#226semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×D5×#2262
C4×D5×SL(2,𝔽3)2
C4⋊(C5⋊#226)2
C5×C4⋊#2262
#117515972
D5×#4623
D5×C4⋊Dic34
C2×C4⋊#2265